What entity satisfies the operation 2-squared? Wait, nothing does? hmmm…
Numbers are a great example of how the demonstration for the existence of a non-observable abstract entity works. We can show that numbers exist through a process of elimination of the alternatives. Numbers can’t be physical objects, nor mental concepts, nor linguistic entities such as “labels.” So numbers can only be numbers. So they exist.
Notice that the operation 2-squared admits of **one and only one **answer to make it true, so the answer can’t be 3, 5, 1, but only 4. It is not a formula like x-squared which can admit of many possible answers. This is an important observation to make.
First, Anti has said in one place that numbers are labels for concepts. So the label “2” =
2. Is that right? No. He confuses the signifier for the concept signified, the term “2” for the concept
2.
Second, he says number-concepts are “derived from physical realities”–presumably, empirical abstractions from observables. He is right the number 2 is an abstraction, but he is wrong to think it is an abstraction
from experience, since the concept of the number 2 is not directly observed anywhere in my field of experience like I can observe the color red which I can extract from several instances of red things to form a concept by which I can apply this very concept to several tokens of the same color. So how does this abstraction of the concept of 2 arise if I don’t observe it? When I see two objects before me, what makes them two objects rather than three objects, and how do I know this? Experience just presents itself to me as a random flux of fleeting and disorganized sensations without structure. The activity of the human mind is what imposes this organization and structure onto sense-experience–neuroscience and psychology have shown this numerous times. So there is nothing in the experience itself which organizes itself into the conception of two objects–rather, the abstraction requires the activity of my mind. So it can only be the innate spontaneous activity of the mind which applies the innate concept of 2 onto my disorganized experience making it possible for me to recognize 2 objects in my experience rather than three. The concept is innate and cannot be derived from experience, but is presupposed in all experience and makes it possible for me to perform the operation of addition on the multitude of objects presented to me.
Take my question toward AntiTheist’s view above. What satisfies the operation 2-squared? Is it physical objects? Perhaps 4 oranges? 4 apples? or maybe 4 people? Or do all of these objects satisfy the operation at once?
So what is the answer? It can’t be each* set *of four things–4 apples, 4 oranges, or 4 people–that satisfies the operation at once, because then we would have four individuals from each set, making, eight, twelve, or infinitely many things. And then the operation 2-squared=8, 12, and so on… would be false.
So suppose it is
just the four physical oranges that satisfies this operation. But then suppose we smash these oranges to many tiny bits, so that 2-squared=25, 34, orangey tiny bits. Then the operation is false. So it is absurd to suppose that the truth of the operation is entirely dependent on the contingent existence of those four oranges. Moreover, if it can
only be the oranges that satisfy the operation, then the 4 apples cannot satisfy it because we stipulated that it is the physical oranges, not the physical apples that satisfies it. But surely we can apply the operation 2-squared to multiple kinds of entities, yes? So what is it, really, that is satisfying this operation which admits of one and only one answer or value?
Maybe it is the
concept of 4, that satisfies the operation. But wait, aren’t concepts mind-
dependent abstract entities (according to Anti)? If they are, then concepts cease to exist when the human mind ceases to exist. Suppose John dies and ceases to exist altogether. Then the concept of 4 whose existence was dependent John’s mind, also ceases to exist with him. Therefore, there is no longer a value that the operation designates and it is therefore false because “2-squared=nothing” is false. But surely the operation 2-squared=4 is still true when John dies. The truth of the operation is certainly not dependent on
John’s own existence. That’s just as absurd as the truth of the operation depending on the contingent existence of four oranges.
So mental concepts and physical objects don’t uniquely satisfy the operation 2-squared.
Therefore, the only entity I can think of that satisfies it is 4. So the number 4 exists. If it did not exist, the operation designates nothing, admits of no answer, and is either meaningless or false.